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DEPARTMENT OF MECHANICAL ENGINEERING

COIJ_F_GE OF ENGINEERING & TECHNOLOGY

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23529

STUDIES ON NONEQUILIBRIUM PHENOMENA INSUPERSONIC CHEMICALLY REACTING FLOWS

By

Rajnish Chandrasekhar, Graduate Research Assistant

and

S.N. Tiwari, Principal Investigator

Progress ReportFor the period ended August 31, 1993

Prepared forNational Aeronautics and Space AdministrationLangley Research Center

Hampton, VA 23681-0001

UnderResearch Grant NAG-I-423

Drs. A. Kumar and J.P. Drummond, Technical Monitors

FLDMD-Theoretical Flow Physics Branch

Submitted by theOld Dominion University Research FoundationP.O. Box 6369

Norfolk, VA 23508-0369

November 1993

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DEPARTMENT OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING & TECHNOLOGY

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23529

i

STUDIES ON NONEQUILIBRIUM PHENOMENA INSUPERSONIC CHEMICALLY REACTING FLOWS

By

Rajnish Chandrasekhar, Graduate Research Assistant

and

S.N. Tiwari, Principal Investigator

Progress ReportFor the period ended August 31, 1993

Prepared forNational Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23681-0001

L{

Under

Research Grant NAG-I-423Drs. A. Kumar and J.P. Drummond, Technical Monitors

FLDMD-Theoretical Flow Physics Branch

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iii iii,,i,i,, iiii ,illi, November 1993

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ABSTRACT

STUDIES ON NONEQUILIBRIUM PHENOMENA IN

SUPERSONIC CHEMICALLY REACTING FLOWS

This study deals with a systematic investigation of nonequilibrium processes in supersonic

combustion. The two-dimensional, elliptic Navier-Stokes equations are used to investigate

supersonic flows with nonequilibrium chemistry and thermodynamics, coupled with radiation,

for hydrogen-air systems. The explicit, unsplit MacCormack finite-difference scheme is used to

advance the governing equations in time, until convergence is achieved.

For a basic understanding of the flow physics, premixed flows undergoing finite rate chemical

reactions are investigated. Results obtained for specific conditions indicate that the radiative

interactions vary substantially, depending on reactions involving HO2 and NO species, and that

this can have a noticeable influence on the flowfield.

The second part of this study deals with premixed reacting flows under thermal nonequilib-

rium conditions. Here, the critical problem is coupling of the vibrational relaxation process with

the radiative heat transfer. The specific problem considered is a premixed expanding flow in a

supersonic nozzle. Results indicate the presence of nonequilibrium conditions in the expansion

region of the nozzle. This results in reduction of the radiative interactions in the flowfield.

Next, the present study focuses on investigation of non-premixed flows under chemical

nonequilibrium conditions. In this case, the main problem is the coupled turbulence-chemistry

interaction. The resulting formulation is validated by comparison with experimental data on

reacting supersonic coflowing jets. Results indicate that the effect of heat release is to lower the

turbulent shear stress and the mean density. The last part of this study proposes a new theoretical

formulation for the coupled turbulence-radiation interactions. Results obtained for the coflowing

jets experiment indicate that the effect of turbulence is to enhance the radiative interactions.

- i

ACKNOWLEDGEMENTS

i,

]i •

1

This is a progress report on the research project, "Analysis and Computation of Intemal Flow

Field in a Scramjet Engine," for the period ended August 31, 1993. Special attention during

this period was directed to "Studies on Nonequilibrium Phenomena in Supersonic Chemically

Reacting Flows."

The authors are indebted to Drs. R. L. Ash, A. O. Demuren, J. P. Drummond, A. Kumar,

and J. Tweed for many useful and constructive suggestions during the course of this study and

in the preparation of the final manuscript.

This work, in part, was supported by the NASA Langley Research Center (Theoretical Flow

Physics Branch of the Fluid Mechanics Division) through the grant NAG-I-423. Th grant was

monitored by Drs. A. Kumar and J. P. Drummond. The work, in part, was also supported

by the Old Dominion University's ICAM Project through NASA grant NAG-I-363; this grant

was monitored by Mr. Robert L. Yang, Assistant University Affairs Officer, NASA Langley

Research Center, Hampton, Virginia 23681-0001.

i _

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TABLE OF CONTENTS

°o°

ACKNOWLEDGEMENTS ................................. 111

HST OF TABLES ...................................... vi

LIST OF FIGURES ..................................... vii

LIST OF SYMBOLS .................................... x

i

L

Chapter

1.

2.

OVERVIEW AND RATIONALE ........................ 1

PREMIXED CHEMICAL NONEQUILIBRIUM FLOWS .......... 5

2.1 Introduction ................................ 5

2.2

2.3

2.4

2.1.1 Literature Survey ........................ 5

2.1.2 Discussion of Physical Model ................ 7i

Basic Governing Equations ....................... 7

Clmmistry and Thermodynanlic Models ................ 10

Radiative Interactions ........................... 13

r

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2.5 Method of Solution ............................ 15

iv

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i,

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I

3. PREMIXED THERMOCHEMICAL NONEQUILIBRIUM FLOWS .... 18

3. ! Introduction ................................ ! 8

3.2 Governing Equations ............................ 22

3.3 Thermal Nonequilibrium ......................... 24

3.4 Radiative Interactions ........................... 25

3.4.1 Local Thermodynamic Equilibrium ............. 26

3.4.2 Non-local Thermodynamic Equilibrium ........... 26

3.5 Method of Solution ............................ 27

4. NONPREMIXED NONEQUILIBRIUM FLOWS ............... 29

4.1 Introduction ................................ 29

4.2 Basic Governing Equations ....................... 32

4.3 Reynolds Stress Turbulence Model ................... 37

4.4 Radiative Interactions ........................... 39

4.5 Method of Solution ............................ 40

5. RESULTS AND DISCUSSION ......................... 41

5.1 Chemical Nonequilibrium and Radiative Interactions ....... . . 41

I

5.2 Therrnochemical Nonequilibrium and Radiative Interactions .... 66

5.3 Turbulence-Chemistry Interactions ................... 84

5.4 Turbulence-Radiation Interactions ................... 113

6. CONCLI JSIONS ................................. I 17

REFERENCES. ....................................... 119

! •

r

Table

2.1

4.1

4.2

LIST OF TABLES

Hydrogen-Air Mechanisnl ........................... 12

Conditions for the Beach experiment .................... 34

Conditions for the Jarrett-Pitz experiment ................. 34

c

i _

T

vi

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!=

ii

!; i

Figure

2.1

2.2

3.1

3.2

4.1

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

5.10

5.10

5.11

5.11

5.11

LIST OF FIGURES

Physical model for premixed chemical nonequilibrium .......... 8

Plane radiating layer between parallel boundaries ............. 14

Schematic diagram of nozzle for air chemistry ............... 21

Schematic diagram of nozzle for lt2--air chemistry ............ 23

Schematic of the coaxial .jet flowfield .................... 33

Temperature profiles .............................. 43

Pressure profiles (y=0.13 cm.) ........................ 44

H20 mass fraction profiles .......................... 45

Otl mass fraction profiles ........................... 46

Grid sensitivity results ............................. 47

Effect of reaction rates ............................. 49

Effect of NO and t102 reactions ....................... 50

I

Profiles of streamwise radiative flux ..................... 52

Profiles of normal radiative flux ....................... 53

Radiation effects on temperature profiles (2-step model) ......... 54

Radiation effects on temperature profiles (18-step model) ......... 55

Radiation effects on temperatnre profiles (35-step model) ......... 56

Radiation effects oll pressure profiles at y = 0.13 cm. (2-step model).. 57

Radiation effects on pressure profiles at y = 0.13 cm. (! 8-step model) . 58

Radiation effects on pressure profiles at y = 0.13 cm. (35-step model) . 59

vii

/

Ii__

5.12

5.12

5:12

5.13

5.13

5.13

5.14

5.15

5.15

5.16

5.16

5.16

5.17

5.17

5.18

5.18

5.18

5.19

5.19

5.19

5.20

5.20

5.20

5.21

Radiation effects on [t20 profiles (2-step model) ............. 60

Radiation effects on tt20 profiles (18-step model) ............. 61

Radiation effects on H20 profiles (35-step model) ............. 62

Radiation effects o11 OI1 profiles (2-step model) .............. 63

Radiation effects on OH profiles (18-step model) ............. 64

Radiation effects o11 Oti profiels (35-model) ................ 65

Air chemistry temperature profiles (l-l)) ................... 68

Temperature profiles (1 -I)) .......................... 69

Pressure profiles (l-D) ............................. 70

Grid resolution study on streamwise radiative flux -- at wall

(j = jmax-I ) . .......... 72

Streamwise radiative flux profiles -- midway (i = jmid) ......... 73

Streamwise radiative flux profiles --wall (i = jmax-1) .......... 74

Normal radiative flux profiles (i = jmid) .................. 75

Normal radiative flux profiles (i = jmax-l) ................. 76

Translational temperatures (i = 1) ...................... 78

Translational temperature (i = jrnid) ..................... 79

Translational temperatures (i = jmax- !) ....... _ ........... 80

Pressure profiles (i = 1) ............................ 81

Pressure profiles (i = jrnid) .......... ' ................ 82

Pressure profiles (i = jmax-1) ......................... 83

Vibrational temperature (j = 1) .................... 85

Vibrational temperatures (i = jmid) ...................... 86

Vibrational temperatures (i = jmax-1) .................... 87

Water mass fraction profiles (j = 1) ..................... 88

viii

i •

i

ili'

i:

5.21

5.21

5.22

5.23

5.23

5.24

5.25

5.26

5.27

5.27

5.27

5.28

5.29

5,30

5.31

5.32

5.33

5.34

5.35

5.36

5.37

Water mass fraction profiles (j = jmid) ................... 89

Water mass fraction profiles (j = jmax-l) .................. 90

Radial profile of initial temperature ..................... 92

Grid resolution study on mean density .................... 93

Profiles of major species concentrations ................... 94

Profiles of mixture fiaction .......................... 96

Profiles of minor species fraction ....................... 97

Radial profile of density ............................ 98

Radial profile of normalized turbulent shear stress ............. 99

Effect of different B.C.'s on turbulent shear stress ............ 100

Mixture fraction profile -- effect of different shear stress B.C.'s . . . 101

Radial profile of mixture fraction ...................... 103

Radial profile of temperature ........................ 105

Profile of water mass fraction ........................ 106

Radial profile of normalized turbulent shear stress ............ 107

Profile of R.It.S. of _?_---_';equation ..................... 108

Effect of "pressure-strain" model on turbulent shear stress ....... 109

Effect of"pressure-strain'" model on temperature . ! .......... 110

Effect of multivariate species PDF on water mass fraction ....... ! 12

Profiles of streamwise radiative flux -- ,larret-Pitz experiment ..... 114

_ • oEffect of turbulent/radmtmn" coupling on temperature ......... 116

ix

i;:

ii: i

LIST OF SYMBOLS

! ,

L

A

Ao

c i

Co

Cp

c

D

E

e_,

H

h

h

k

kb

kf

k

Mt

P

band absorptance, m "!

band width parameter, m -1

concentration of the jth species, kg-molehn 3

correlation parameter, (N/m2) -Im-I

constant pressure specific heat, J/kg-K

speed of light

binary mass diffusivity

total vibration energy

Planck's fimction

mass fraction of the jlh species

total enthalpy, J/kg

static enthalpy, J/kg

Planck's constant = 6.6262× 10 -27 erg.sec

turbulent kinetic energv = ,/._t'./9

backward rate constant

forward rate constant

Boltzmann's constant = 1.3806× 10 -16 erg K -I

turbulent Mach number

pressure, N/m 2

C ¸

I_ I

i _.

j •r,

% "

Pi

qa

R

S

H, V

w i =

x,y =

partial pressure of the jth species

total radiative flux

gas constant

integrated band intensity. (N/m2)lm -2

translational-rotational temperature, K

vibrational temperature, K

velocity in x- and y- directions, m/s

production rate of the jth species, kg/m3-s

physical coordinates

Greek Symbols

t_ -----

0v =

top =

,\ =

it =

_, 71 =

p =

T =

6 =

ratio of specific heats

turbulent dissipation rate

characteristic vibrational temperature

Planck mean absorption coefficient

second coefficient of viscosity, wavelength

dynamic viscosity, kg/m-s

computational coordinates

density

Stefan-Boltzmann constant = 5.668×10 -8 erg K -4

shear stress

equivalence ratio

turbulent Prandtl or Schmidt number

thermal conductivity

wave number, m -I

Fxi

i_i_

i

k

Superscripts

Favre -- averaging (density -- weighted)

fluctuating density -- weighted variable

Reynolds -- averaging (time -- weighted)

fluctuating tirne -- weighted variable

I

,

xii

!, ,.

1 ¸

t ?

I'

Chapter !

OVERVIEW AND RATIONALE

In recent years there has been a renewed interest in the development of a hypersonic

transatmospheric aerospace vehicle capable of flying at sub-orbital speeds. A hydrogen-

fueled supersonic combustion rainier (scramjet) engine is a strong candidate for propelling

such a vehicle. The airflow is compressed inside the engine inlet and the supersonic

combustion takes place inside the scramjet combustor. After this, the burned gases

are expanded through the nozzles, followed by the undersurface of the vehicle. For

a better understanding of the complex flowfield in different regions of the engine, both

experimental and computational techniques have been employed. The complexity of these

flows makes traditional wind tunnel tests quite difficult, ttowever advances in computer

architecture and efficient algorithms, make it possible to numerically investigate the flow

in various sections of the scramjet module. Therefore, Computational Fluid Dynamics

(CFD) is an extremely valuable tool for numerical simulation of supersonic combustion.

The flowfield in the combustor and nozzle sections of the scramjet is characterized

by very short residence times. This could lead to chemical nonequilibrium, since the

chemical reaction time will be of the same order of magnitude as the flow residence

time. Furthermore, the scramjet flowfield is characterized by diffusive or non-premixed

burning. However, premixed flows can be considered for preliminary studies, as well as

for flow analysis in detonation wave engines. This lays the foundation for the present

work, which is carried out in a systemalic manner. .

tj ,

h

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!

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2

In the case of premixed flows, the air and fuel are thoroughly mixed in the required

stoichiometric proportion, prior to combustion, klncertainty in the choice of appropriate

chemistry models can affect predictions of ignition delay. This underlines the need

to study effects of chemical kinetics. Furthermore, this high enthalpy gaseous mixture

encounters very short residence times and wide temperature and pressure variations. This

could cause thermochemical nonequilibrium conditions. I11 addition, the combustion of

hydrogen and air in the scran_jet combustor results in absorbing-emitting gases such as

water vapor and hydroxyl radicals. Existence of such gases makes it necessary to include

the effect of radiation heat transfer.

The objective of the first part of this study is to investigate premixed flows under-

going finite rate chemical reactions, ltere, the key problem is to determine the impact

of chemical kinetics on the radiative interactions. The specific problem considered is

the premixed flow in a channel with a ten-degree compression ramp. Three different

chemistry models are used, accounting for increasing numbers of reactions and partici-

pating species. Two models assume nitrogen as inert, while the third chemistry model

accounts for nitrogen reactions and NOx formation. The tangent slab approximation is

used in the radiative flux formulation. A pseudo-gray model is used to represent the

absorption-emission characteristics of the participating species. R_sults obtained for spe-

cific conditions indicate that the radiative interactions vary substantially, depending on

reactions involving ttO2 and NO species, and that this can have a noticeable influence

on the flowfield. This provides the analytical tools required for further investigation of

nonequilibrium processes in supersonic combustion.

The objective of the second part of this study is to investigate premixed reacting

flows under thermal nonequilibrium conditions, ltere, the critical problem is coupling of

vibrational relaxation process with the radiative heat transfer. This has been implemented

i "

!!,

i,•!!! •

3

in a unique manner. The specific problem considered is the premixed expanding flow in

a supersonic nozzle. The reacting flow consists of seven species, one of which is the inert

N2 molecule. The thermal state of the gas is modeled with one translational-rotational

temperature and five vibrational temperatures. The harmonic oscillator model is used

in the formulation for vibrational relaxation. Results obtained for this case indicate the

presence of nonequilibrium in the expansion region. This, in turn, reduces the radiative

interactions and can have a significant influence on the flowfield.

In the case of non-premixed flows, the engine efficiency is strongly influenced by

turbulent mixing of the filel and oxidizer and its effect on chemical reactions. A variety of

turbulence models can be applied to the analysis of the scrarnjet flowfield. Most of these

models assume the turbulence to be isotropic. However, the occurrence of heat release due

to chemical reactions, and the presence of shocks, could lead to anisotropic turbulence.

This effect can be simulated by the Reynolds Stress models. Consequently, the objective

of the third part of this study is to investigate non-premixed flows undergoing chemical

nonequilibrium. A differential Reynolds Stress turbulence model has been applied to the

Favre-averaged Navier-Stokes equations. An assumed Beta Probability Density Function

is applied to account for the chemical source terms in the conservation equations. The

resulting formulation is validated by comparison with experimental data on reacting

supersonic coflowing .jets. Results obtained for specific conditions demonstrate that the

effect of chemical reaction on the turbulence is significant.

As stated earlier, the combustion of hydrogen and air in the scramjet combustor re-

suits in gases such as water vapor and hydroxyl radicals. Occurrence of such absorbing-

emitling gases in the turbulent flame, implies the need to simulate the effect of radiative

interactions. Thus, the objective of the last part of this work is to investigate turbulent

i .'_ _

L

4

radiating flames, tlere, the unsolved problem is the coupled turbulence-radiation inter-

action, for which a novel theoretical formulation has been proposed. Results obtained

for the coflowing jets experiment indicate that the effect of turbulence is to enhance the

radiative interactions.

The explicit, unsplit MacCormack finite-difference schenle is used to advance the

governing equations in time, until convergence is achieved. The chemistry source term

in the species equation is treated implicitly to alleviate the stiffness associated with fast

reactions. Details of the theoretical formulations along with the methods of solution,

are given in the ensuing chapters. Premixed flows are discussed in Chaps. 2 and 3.

Chemical nonequilibrium is dealt with in Chap. 2, while thermochemical nonequilibrium

receives attention in Chap. 3. An expos_ of non-premixed flows is given in Chap. 4. 3lhe

computational results obtained are presented and discussed in Chap. 5.

i ,

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Chapter 2

PREMIXED CHEMICAL NONEQUILIBRIUM FLOWS

in this chapter, the chemical nonequilibrium phenomena pertaining to premixed,

supersonic reacting flows are discussed. First, the problem is introduced. Then, the

relevant literature is reviewed and the physical model is described. This is followed by

a layout of the governing equations. After this, the method of solution is discussed.

2.1 Introduction

!,

! .

In tile case of premixed flows, the air and the fuel are thoroughly mixed in the required

stoichiometric proportion, prior to combustion. This gaseous mixture encounters very

short residence times (0[1.0 msec]), which will be of the same order of magnitude as the

chemical reaction time. Uncertainty in the choice of appropriate chemistry models can

affect predictions of ignition delay. This underlines the need to study effects of chemical

kinetics. In addition, the combustion of hydrogen and air in the scramjet combustor

results in absorbing-emitting gases such as water vapor and hydroKyl radicals. Existence

of such gases makes it necessary to include the effect of radiation heat transfer.

2.1.1 Literature Survey

• i

7

Several computer programs have been developed and applied to gain more insight

into the problem involving a scramjet flowfield II-3]*. Kumar [1] carried out numerical

simulations of scranliet inlet flowfields. Drummond et al. [2, 3] developed a spectral

method code for predicting the behavior of supersonic reacting mixing layers.

* Numbers in brackets indicate references.

L_ ?

if'

(!

6

Considerable work has been done in the past decade to model the chemical kfl_etic

mechanism of the hydrogen-air system [4-7]. A simple two-step finite-rate chemistry

model was used by Chitsomboon et al. [4] as well as by Rogers and Schexnayder [5].

A complete model wonld involve some 60 reaction paths, making numerical solution

very difficult, if not impossible [5]. The two-step model [4, 5] has only font species

and two reaction paths, and is used for preliminary studies, ltowever, there are several

limitations to this model, such as ignition-phase inaccuracy (i.e. a much shorter ignition

delay), and overprediction of flame temperature as well as longer reaction times. Recent

improvements in this area include an 8-species, 14-reaction model used by Shuen and

Yoon [6]. While none of these aforementioned models account for nitrogen reactions (by

assuming nitrogen was inert), recent developments in this area include a 15-species,

35-reaction model reported by Carpenter [7]. This latter model accounts for NOx

formation and other nitrogen reactions in the hydrogen-air system.

There are several models available in the literature to represent the absorption-

emission characteristics of molecular gases [8-13]. Sparrow and Cess [8] wrote a

definitive text on radiative heat transfer. Tien [9] as well as Cess and Tiwari [10]

investigated thermal radiation properties of gases. Band models for infrared radiation

I

were reviewed by Edwards [1 I] and Tiwari [12, 13].

One- and two-dimensional radiative heat transfer equations for various flow and

combustion related problems are available [14-22].

dimensional radiative transfer. Chung and Kim

Tsai and Chan [14] studied multi-

[15] reported a solution for two-

dimensional radiation using the finite element method. Coupled radiation and convection

were investigated by lm and Ahluwalia [16] as well as Soufiani and Taine [17]. Tiwari

[18, 19] studied transient radiative interactions in gases, Mani et al. [20-22] obtained

numerical solutions of supersonic chemically reacting and radiating flows.

7

As the above literature survey clearly indicates, there is a need for additional studies

in this key field. The objective of this chapter is to determine the impact of chemical

kinetics on the radiative interactions. The specific problem considered is tile premixed

flow in a channel with a ten-degree compression ramp. Three different chemistry models

are used, accounting for increasing number of reactions and participating species. Two

models assume nitrogen as inert, while the third chemistry model accounts for nitrogen

reactions and NOx formation. The tangent slab approximation is used in the radiative

flux formulation. A pseudo-gray model is used to represent the absorption-emission

characteristics of the participating species.

2.1.2 Discussion of Physical Model

The specific problem considered is the supersonic flow of premixed hydrogen and

air (stoichiometric equivalence ratio & = 1.0) in a channel with a compression corner on

the lower bonndary (Fig. 2.1). The physical dimensions considered for obtaining results

are L = 2cm., X I = 1 cm., X2 = 2 cm., Lx = X i + X2 = 3 cm., and c_ = I0 degrees. The

flow is ignited by the shock from the compression corner. The inlet conditions which

are representative of scramjet operating conditions, are P_ = 1.0 atm., 'I'_ -- 900 K and

M,-_ = 4.0. This same flow has been computed by several researchers [4, 6, 20-22] as

a benchmark case.

2.2 Basic Governing Equations

:i

I

The physical model for analyzing the flowfield in a supersonic combustor is described

by the Navier-Stokes and species continuity equations. For two-dimensional flows, these

equations are expressed in physical coordinates as [2, 3]

0U O/:' 0(, _

+ + + u : 0 (2.1)

i__'

LY

L X ,--

M=4T=900 K

P=I atm

O_=10°

L=2cm

Lx = 3 cm

Xl=lCm

X 2 = 2cm

i •

Fig. 2.1

X 1 _ X 2 -

Physical model for premixed chemical nonequilibrium

i:

L:

i •

If

,j

where vectors U, F, G' and It are written as,

G

H

= pt',

p E

P.[j

pt!,)

pu ° + p + T:,.:_

puv + Ty_l

(pE + p)u + T_.U + T_._P -t- q,.:,, at. qlt._

p,,.rj- el)_pt'

plll_ -_- Ty:r

pv 2 + i ) + r,._

(pE -t- p)_' + T_.FI + Dye' + q,w + qlhl

P",f.i - pl)_

0

0

0

0

-- t t.'j

The viscous stress tensors in the I;' and G' terms are given as,

9

(2.2)

i •

}k

i'

- 2if-'-'" = - \_ + a:/) a.',_

T,,,,_- -,,,\_ + a:_j - 2_,_

(2.3a)

(2.3b)

(2.3c)

where .,k = _2ff invokes the Stokes' hypothesis. This assumption of zero bulk viscosity

is true for an incompressible gas, as well as for boundary layer type flows [23-25]. The

quantities q,_ and q,_l in the 1;' and (; terms are the components of the conduction heat

flux and are expressed as

q(':F07' 01' (2.4)

-- ('_ ; q'_' = -+('OT

The molecular viscosity ii is evaluated fiom Sutherland's formula [22].

internal energy E in Eq. (2.2) is given by

10

The total

p It 2 -'1- Z;2 m.

z - + + _ 1,,f_- (2.5)p 2 .i= i

Specific relations are needed for the chemistry and radiative flux terms.

discussed in the following sections.

These are

2.3 Chemistry and Thermodynamic Models

Cllemical reaction rate expressions are usually determined by summing the contribu-

tions from each relevant reaction patll to obtain tile total rate of change of each species.

Each path is governed by a law of mass action expression in which the rate constants

can be determined front a temperature dependent Arrhenius expression. The reaction

mechanism is expressed in a general fimll as

H8 71.q H

.i=I .i=1

, i = 1,7_7" (2.6)

where ns = number of species and 7;;" = number of reactions. The chemistry source

terms _b.i in Eq. (2.2) are obtained, on a mass basis, by multiplying the molar changes

and corresponding molecular weight as

,,, )[_i'.i = /_tiC'i.. : kti. _ ('Yi.i - "_;.i t'Y;i:1

71 S tt -1

H ('_"''/, .j = 1,,._- k/,i -',1, j

I

H ("_i,."" 711

717=1

(2.7)

The reaction rate constants _:fi and l,'t,i, appearing in Eqs. (2.6) and (2.7), are deter-

mined from an Arrhenius rate expression as

.... t, , { _;'"_ (2.8): ,";'

ii

Table 2.1 Hydrogen-Air Mechanism

12

I .

i.

ii/

i!,

i:

i

(!)(2)(3)(4)(5)(6)

(7)

(8)

(9)

(lo)(ll)(12)(13)(14)(15)(16)

(17)(18)(19)

(20)(21)(22)(23)(24)

(25)(26)

(27)

(28)(29)

(30)(31)(32)

(33)

(34)

(35)

(1")

(2")

REACTION

02 + t!2 "_ Oil + OH

02 + t1¢_ Ott+O

H2 +Oil # !120+H

tt2 +O ¢_ Olt+O

Ott + Ot-I ¢_ tt20 + O

tt+O1t+M # t120+M

H+ H+M _ !12 +M

tt +O2 4- M _ ttO2 + M

Oil + [IO2 # 02 + tt20

tt +[IO2 '_ !t2 + 02

It + I-IO2 _ OH + OH

O+ 1tO2 ff 02 +Oit

1102 + 1tO2 _ 02 + 11202

t12 + !tO2 _, "11+ 11202

O11 + 11202 <#.1-1204- 1102

II + 11202 4_, 1120 -1-O11

O + 11202 _,, O11 + 1102

11202 + M _ O11 +OH + M

02 +M ,_ O+O+M

N2=M_N+N+M

N+O2 <=_,O+NO

N +NO ¢_, O+N2

N+Ott _ H+NO

H +NO+M _ I INO+ M

H + tlNO 4:¢ ti2 + NO

O + ItNO ¢4, Oil + NO

Oti + 11NO _ It20 + NO

ttO2 + HNO ¢¢, 11202 + NO

ttO2 + NO _ O!! + NO2

H +NO2 _ OtI+N0

O +NO2 <_,02 +NO

NO2 +M <=_O+NO+ M

M+OH+NO _" !INO2 4-M

M+OH+NO2 _ I INO3 + M

Oil + IINO2 _ 1120-I-NO2

global

1-12+ 02 _ 2011

2011 + 112¢_" 21120

A(moles)

1.70x 1013

!.42x 1014

3.16x 1007

2.07x 1014

5.50x 10 t3

2.21 x !022

6.53x 1017

3.20x 10 lr

5.00)< 10 I-_

2.53× 1013

1.99× I0 u

5.00× 1013

1.99X 1012

3.01×I0 Iv

!.02× 1013

5.00× 10 TM

!.99X 10 t3

1.21 × 1017

2.75× 1019

3.70× 102t

6.40× 1009

i.60x 1013

6.30× 1011

5.40× 10 v5

4.80× 1012

5.00x 0II

3.60× 013

2.00x 0 t2

3.43× 012

3.50x 0 TM

1.OOx 0 t3

1.16×1016

5.60× 10 v5

3.00x 1015

i.60× 10 t2

2-slep

1! .4 x I 047

2.50 x 106"I

N(cm 3)0

0

1.8

0

0

-2.0

-! .0

-I .0

0

0

0

0

0

0

0

0

0

0

-I .0

-1.6

1.0

0

0.5

0

0

0.5

0

0

0

0

0

0

0

0

0

model

-10.0

: -13.0

E(calories/gm-mole)

48150

16400

3O30

! 3750

7000

0

0

0

1000

7O0

1800

1000

0

18700

1900

10000

5900

45500

118700

225000

6300

0

0

-600

0

0

0I

0

-260

1500

600

66000

-1700

-3800

0

4865

42500

I.

i r

i

i

}, .

i

t

i[

i .

i

r.

t

it •

2.4 Radiative Interactions

13

Evaluation of the energy equation presented in Eq. (2.2) requires an appropriate

expression for the radiative flux term, qR. Therefore, a suitable radiative transport model

is needed• Various models are available in the literature to represent the absorption-

emission characteristics of the molecular species [22]. The equations of radiative transport

are expressed generally in integro-differential forms. The integration involves both

the frequency spectrum and physical coordinates. In many realistic three-dimensional

physical problems, the complexity of the radiative transport equations can be reduced by

introduction of the tangent-slab approximation. This approximation treats the gas layer

as a one-dimensional slab in evaluation of the radiative flux (Fig. 2.2).

Detailed derivations of radiative flux equations for gray as well as nongray radiation

have been carried out previously I22]. For a multiband gaseous system, the nongray

radiative flux in the normal direction is expressed as

qR(!¢) = cl - c'2 + ___ Aoi .i=1 •

- [3 _'o, ._)ldz- J

4- [ a.: - :/) a: , .(2.15)

Information on the band absorptance ,.hi and other quantities are available in the cited

references.

For a gray medium, the spectral absorption coefficient t¢_, is'independent of the wave

number, and an expression for the radiative flux is obtained as [8, 22]

_ 2 tzdz

" } (2.16)

14

J

r

! ,

r

! .

Fig. 2.2 Plane radiating layer between parallel boundaries

! ,

i'

i,

i

F

i '

I •

i:

15

It is computationally more efficient to use Eq. (2.16) in the general energy equation

than Eq. (2.15). This is because by differentiating Eq. (2.16) twice (using the Leibnitz

formula) the integrals are eliminated and the following inllomogeneous ordinary differ-

ential equation is obtained:

1 d2qR(y) 9 3 de(y)

_,:-" d._/-' ,1qn(7.1) - _,. d!/ (2.17)

The solution of Eq. (2.17) requires two boundary conditions which are given for non-

black diffuse surfaces as 18]

( ) '1 I [qn(;z)].=o L_J = 0 (2.18a);] 2 " :_" .=o

,-2 [qn(:/)],,=r,+ :_,-SLa,:/J,=r.

For black surfaces, the emissivity c_ = (2 = 1 and Eqs. (2.18) reduce to simpler forms.

An appropriate model for a gray gas absorption coefficient is required in Eqs. (2.16)-

(2.18). This is represented by the Planck mean absorption coefficient, which is expressed

for a multi-band system as [8, 22]

PJ _ (;z'),_'_(7') (2.19)-- Cwi

t; = t;1" c_T.l(!l ) =

I

It should be noted that top is a function of the temperature and the partial pressures Pj

of the species.

2.5 Method of Solution

The flowfield in the combustor is represented by the Navier-Stokes equations and

by the appropriate species continuity equations. The solution scheme is based on the

SPARK code [2-4, 7. 20-22] available at NASA Langley Research Center. The finite-

difference method using the explicit, unsplit MacCormack scheme [26] is used to solve

i"

J

.

]

r"

;

16

the governing equations. Incorporation of the finite-rate chemistry models into the fluid

dynamic equations can create a set of stiff differential equations. Stiffness is due to a

disparity in the time scales of the governing equations. In the time accurate solution, after

the fast transients have decayed and the solutions are changing slowly, taking a larger

time step is more efficient. But explicit methods still require small time steps to maintain

stability. One way around this problem is to use a fully implicit method. However, this

requires the inversion of a block multi-diagonal system of algebraic equations, which is

also computationally expensive. The use of a semi-implicit technique [27, 28] provides

an alternative to the above problems. This method treats the source term (which is the

cause of the stiffness) implicitly, and solves the remaining terms explicitly.

The governing equations are transformed from the physical domain (:r, ._/) to a

computational domain (_, 71), using an algebraic grid generation technique similar to the

one used by Smith and Weigel [29]. In the computational domain, Eq. (I) is expressed

as [30]

where

of aOat 4 4- H = 0 (2.20)

i /

/_ = I;.I, [" = 1:'_1,1- (-;'_',1

(_ = (;:rE- !;'_/_ , H = It.I

(2.21 )

! ,

fi•

[

Once the temporal discretization has been performed, the resulting system is spatially

differenced using the explicit, unsplit MacCormack predictor-corrector scheme [26]. This

results in a spatially and temporally discrete, simultaneous system of equations at each

I

i •

!,

17

grid point. Each simultaneous system is solved, subject to initial and boundary conditions,

by using the Householder technique [31, 32]. At the supersonic inflow boundary,

characteristic flow quantities are specified as freestream conditions. At the supersonic

outflow boundary, non-reflective boundary conditions are used, i.e. flow quantities are

extrapolated from interior grid points.

i, iI

i

Chapter 3

L

i :

i .

t

PREMIXED THERMOCHEMICAL NONEQUILIBRIUM FLOWS

In this chapter, the thermochemical nonequilibrium phenomena pertaining to pre-

mixed, supersonic reacting flows are discussed. First, the relevant literature is reviewed,

and then the physical models used in this study are described. This is followed by a

description of the governing equations. After this, the method of solution is discussed.

3.1 lntroduciion

r ,

As stated in Chap. 2, in the case of premixed flows, the air and the fuel are thor-

oughly mixed in the required stoichiometric proportion, prior to combustion. This high

enthalpy gaseous mixture characterized by very short residence times and wide tempera-

ture and pressure variations. This could lead to thermochemical nonequilibrium. In order

for molecules to dissociate, they must be excited in all three energy states (rotational,

translational and vibrational). After dissociation, the translational and rotational temper-

atures relax towards equilibrium faster than tile vibrational tempei'ature. This makes the

study of vibrational nonequilibrium an important issue. Furthermore, the combustion of

hydrogen and air in a scramjet combustor results in absorbing-emitting gases such as wa-

ter vapor and hydroxyl radicals. Existence of such gases makes it necessary to consider

the effect of radiative heat transfer.

In the presence of a radiation field, if the energy exchange is dominated by a

collisional process, then the conditions of local thermody.namic equilibrium (LTE) exist.

Otherwise, the system is considered to be in the state of non-local thermodynamic

18

I :

i" :

19

equilibrium (non-13"E or NUfE). Further discussion on this is provided in the section

on radiative interactions.

Several theoretical and computational studies on the nonequilibrium flow of air have

been carried out [33-40]. Blythe [34, 35] carried out analytical studies of nonequilibrium

flows in nozzles. Stollery and Park reported one of the first numerical solutions to the

vibrational relaxation problem. Cheng and Lee [37] investigated freezing effects of

chemical nonequilibrium flows in nozzles. A comprehensive review of the literature

prior to 1968 was undertaken by Hall and Treanor [38]. Further numerical results for

thermochemical nonequilibrium were discussed by Anderson [39, 40].

Vibrational relaxation effects are important in mixtures of combusting gases [41-43].

The relaxation rates of some gases were discussed by von Rosenberg et al. [41, 42] and

by Kung and Center [43]. Vibrational relaxation effects are also important in lasers

[44-47]. Kothari et al. [44] obtained numerical simulations for chemical lasers. Gas

dynamic lasers were discussed by Anderson [45], by Reddy and Shanmugasundaram

[46], and by Wada et al. [47].

i•

I" :

r

In recent years, thermochemical nonequilibrium effects in atmospheric re-entry flows

have received considerable attention [48-53]. Rakich et al. [48] studied flows over bluntI

bodies. Lee I149] discussed the basic governing equations. Gnoffo [50] developed the

LAURA code for computing reentry flows. Candler and MacCormack [51] focused on

ionization effects. Gnoffo et al. [52] comprehensively laid out the conservation equations

for thermochemical nonequilibrium. Desideri et al. [.53] discussed benchmark results of

a workshop on hypersonic reentry flows.

The combustion of hydrogen and air in the scramjet combustor results in gases such

as water vapor and hydroxyl radicals. It is known that th_ presence of water vapor gives

I,

I •

i _ '1

i ¸

!IL

r ,

!>

/

20

rise to rapid relaxation rates [41,42, 54, 55]. Finzi et al. [54] and Center and Newton [55]

investigated the vibrational relaxation effects of water vapor. The vibrational relaxation

effects of hydrogen-air combustion have been very briefly touched upon recently by

Grossmann and Cinella [56, 57].

Furthermore, water vapor is an absorbing-emitting gas. Existence of such gases

makes it necessary to include the effect of radiation heat transfer. Coupled radiative

transfer with chemical nonequilibrium has been studied earlier in Chap. 3 of this study.

The effect of vibrational nonequilibrium upon radiative energy transfer in hot gases has

also been investigated [58-61]. Tiwari and Cess [58] discussed a new formulation for

the non-LTE radiation. Goody [59] studied radiation effects in the upper atmosphere.

Coupled radiative and vibrational relaxation were discussed by Gilles and Vincenti [60].

Radiative cooling effects were investigated by Wang [61].

As this literature survey clearly indicates, there is a need for additional investigation

in this critical area. The objective of this chapter is to study the coupled interaction of

vibrational relaxation and radiative heat transfer, in the presence of finite rate chemical

reactions. The thermal state of the gas is modeled using one translational-rotational

temperature and five vibrational temperatures. The harmonic oscillator model is used inI

the formulation for vibrational relaxation. The radiative interactions are investigated in

both streamwise and transverse directions. The tangent slab approximation is used in the

radiative flux formulation. An optically thin assumption is made in the non-LTE model.

Two physical problems are considered for this study. The first one is reacting airflow

in a hypersonic nozzle (Fig. 3.1 ), which is a benchmark case [53] used for code validation.

Inlet reservoir conditions for this flow are l',, = 1.5:t × 10_ l_a,, 7'o = 6500 K.

PO ""

i

1.53x108 Pa

To = 6500K

0.006

I. L = 1.13mr

0.4

LI

Fig. 3.1 Schematic diagram of nozzle for air chemistry

Ii

! •

i

(

iil•

L

22

Based on validation of the code for the problem discussed above, the second physical

problem considered is the supersonic flow of premixed hydrogen and air (stoichiometric

equivalence ratio 4' = 0.3) in an expanding nozzle (Fig. 3.2). The physical dimension

considered for obtaining results is Lx = 2 m. The flow is ignited by the high enthalpy of

the flowfield. The inlet conditions which are representative of scramjet combustor exit

conditions, are I_oo = 0.8046 atm, 7;:,o = 1890 I_" and hlo_ = 1.4. A one-dimensional flow

has been computed by Grossmann and Cinella [56, 57].

3.2 Governing Equations

The Navier-Stokes and species continuity equations used in this study have already

been discussed in Sec. 2.2 of previous Chap. 2. The additional governing equations,

different from Eqs. (2.1) and (2.2), are described as

It =

L pewEq. (2.2)

1pu El,"' .!

Eq. (_.2)k pv E_.i

9 9 1

Eq. (.._)

-- e.j

(3.1)

i

The total internal energy E in Eq. (2.2) is modified as

_) .) ?11 111,

P u" + _,"- + + 1,j/j + bz,,,j (3.2)

p 2 .i= 1 .i= 1

where Ev is the total vibrational energy. Specific relations are needed for the chemistry

and radiative flux terms. These are discussed in the following sections.

7:_. ..... ,.. :-7! : _ T -i - i

Yb/Lx = 0.25 [1 + sin(_ x/4)]

I I

_ -_ ------- -_ X --------.__l

I-- Lx -_I

Pe

r e

Moo = 1.4, Poo = 0.8046 , Too = 1890 K, Lx = 2.0 ml, _= 0.3

"oFig. o._ Schematic diagram of nozzle for H2--aiz chemistry

I,O

L_

•i •

i .i,

24

The chemistry and thermodynamic model used in this study is the same as described

in Sec. 2.3 of Chap. 2. A truncated 7-species, 7-step chemistry model, derived from

Table 2.1, is used in this study. In order to account for the effect of vibrational relaxation

on the chemical reaction rates, the equivalent temperature Tcq,,i,.. in Eqs. (2.8)-(2.10) is

expressed as [62]

'l'rq,,i,,. = k/_. 7'_" (3.3)

!,

3.3 Thermal Nonequilibrium

A silnplified thermodynamic model fi_r the mixture of gases is necessary. Each

species contains translational and rotational energy states in thermodynamic equilibrium

and the vibrational energy is described by a tlarmonic oscillator, which is not in equi-

librium [24, 62]. A Landau-Teller model is used to determine the effect of vibrational

relaxation on the energy production. Furthermore, ionization effects are ignored. It

should be noted that monoatomic species like O and tt are not vibrationally excited.

For the range of temperatures considered in this study, a harmonic oscillator can be

assumed. Accordingly, the vibrational source terms in Eq. (3.1) are expressed as

F'* - El,ci -- Jvi (3.4)

71ci

where

I_iOvi(3.5)

= 1

The asterisk * in Eq. (3.5) denotes the equilibrium value and

h ca.'iO,,i = ----=-- (3.6)

I,'

where i = H2, 02, 1-t20, OH and N2.

ci

¢,

[

25

t.i) isIn Eq. (3.4), the equivalent relaxation time q,-; of a mixture of gases (i =

given by the linear mixture rule [41, 42]

1 _ ./'l + f2 + .13 + .... + fJ (3.7)

7lci 7lci l 71ci2 71ci:l qcij

which accounts for the acceleration of vibrational relaxation observed in the presence of

tt20 [41--43]. Analogy may also be drawn to electrical circuits, with resistors in parallel.

In Eq. (3.7), the local vibrational relaxation time 71,.i.i of a molecular collision pair

(i.j) is given by an empirical correlation [63], curve-fitted from experimental data as

= ,-' :,,IoooJ

-0.OIS/tl.f" ) - 18.,12] (3.8)

where

tt,.1 -- Itii'ttJJ (3.9)Itii + It i.i

In Eq. (3.8), iti.i is the effective molecular weight for a pair of colliding molecules (i,j).

Values of O,,i for N2, 02 and tt20 are obtained from [64] and for t t2 and Oil from [65]

and [66], respectively.

t

3.4 Radiative Interactions

I

Evaluation of the energy E in Eq. (2.2) requires an appropriate expression for the

radiative flux term qR. This radiation is emitted and absorbed by the "photon fluid" [60].

If the photon field is in equilibriunl with the vibrational and translational fields, then

the radiation is said to be of the Local Thermodynamic Equilibrium (LTE) type. This

means that the LTE process is collision dominated. On the other hand, if the photon

field also undergoes nonequilibrium, then the radiation is considered to be of the Non-

local Thermodynamic Equilibrium (NLTE) type. This ilnplies that the NUI'E process is

emission dominated.

i '

ii,

•3.4.1 Local Thermodynamic Equilibrium

26

i

f

The LTE radiative transfer model is the same as discussed in See. 2.4 of Chap. 2 and

so it is not repeated here. This method of coupling the LTE radiation with the governing

equations, is similar to a formulation discussed by Gokcen and Park [67].

3.4.2 Non-local Thermodynamic Equilibrium

The non-UFE radiation model is discussed here. Relevant information on relaxation

processes, nonequilibrium transfer equations and radiative flux equations is provided in

[58]. The basic equations developed can be used to investigate radiative interactions

of gray as well as nongray gases under nonequilibrium conditions. In this study,

however, the nonequlibriurn radiative interactions are considered only in the optically

thin conditions. A brief discussion of applicable equations is provided here.

The nonequilibrium radiative transfer equation

vibrational states may be written as [58-61]

for two level transitions between

dl_,= ,;_,(.1_, - /_,) (3.10)

d.q

where I,_ is the intensity of radiation.

function and is defined as

In Eq. (3.10), .1_ is the nonequilibrium source

I

r +.l,.,., : /3,.,,/ "

L 7/, -F q,_

.'_ = J d[_ j" ,¢,,,l_,da., (3.1 l)

where _ is the solid angle, and B_, is the black-body intensity of radiation. It should

be noted that absorption is an equilibrium process, whereas the nonequilibrium influence

comes only through the emission process (source fimction).

I _ ,

i

!?

27

The time constant 7/,. in Eq. (3.11) is the radiative lifetime of vibrational states, and

this is expressed as [58]

1 _ 8cw- ,':,' (3.12)'/] r

where 77 denotes the number density of the radiating molecules and ¢ represents the

integrated band intensity of a vibration-rotation band.

The influence of nonequilibritun radiation is most apparent in the optically thin limit.

wherein the divergence of the radiative flux can be expressed as [58]

(] ([ R

+3,,c] : C1_,(_)] (3.13)or_ . -"[,/,./

where Ao is the band width parameter and tlo is the nondimensional path length, and these

are defined in the cited references. It can be seen front Eq. (3.13) that the contribution

of the non-LTE (non-local thermodynamic equilibrium) is obtained simply by adding a

correction involving the nonequilibrium parameter ,1_ to the divergence of the radiative7h

Flux.

3.5 Method of Solution

I .¸ ,

!:

]'he method of solution used in this study is the same as discussed in Sec. 2.5 of

I

Chap. 2. Additional details are presented in this section. Only the upper half of the flow

domain is computed, as the flow is assumed to be symmetric about the centerline of a

two-dimensional nozzle. The upper boundary is treated as a solid wall. This implies

a no-slip boundary condition (i.e. zero velocities). The wall temperature and species

mass fractions are extrapolated front interior grid points, by assuming an adiabatic, black

and non-catalytic wall. The pressure is also extrapolated by using the boundary layer

approximations for the pressure gradient. Symmetry boundary conditions are imposed at

the lower boundary. Initial conditions are obtained by specifying freestream conditions

i ,

throughout the flowfield.

convergence is achieved.

28

The resulting set of equations is marched in time. until

The details of the radiative flux formulation and method of

solution are available elsewhere [22].

i "

i

i.?

i

r_

i k ¸¸

! ,

i _ .F

r

f J

il,

Chapter 4

(

I ,

NONPREMIXED NONEQUILIBRIUM FLOWS

In this chapter, the chemical nonequilibrium phenomena pertaining to non-premixed,

supersonic, turbulent reacting flows are discussed. First, the relevant literature is re-

viewed, with a view towards presenting the problem statement and scope, as well as

the physical models used. Then, the basic governing equations are described. This is

followed by a discussion of the method of solution.

4.1 Introduction

i'

i ,

f

J

i'

In tile case of non-premixed flows, engine efficiency is strongly influenced by

turbulent mixing of the filel and oxidizer and its effect on the chemical reactions.

Also, the presence of absorbing-emitting species, like water vapor and hydroxyl radicals,

implies the need to consider radiative heat transfer. The objective of this chapter is to

study the coupled turbulence-chemistry-radiation interactions, in the presence of chemical

nonequilibrium.

Several experimental and computational techniques have been developed towards a

better understanding of the mixing and burning within a supersonic free shear layer or

jet [3, 68]. The shear layer and the single jet simulate the parallel injection of hydrogen

filel in a scramjet engine without introducing complexities arising from the combustor

geometry. Drummond [3] presented numerical solutions for a supersonic reacting mixing

layer. Eklund et al. [68] discussed computational and experimental results for coaxial

reacting jets.

29

r •

i

[

k

r

i "!

j

i :

i "

l •

r ....

30

A variety of turbulence models [69-74] can be applied to the analysis of the scramjet

flowfield. These range from the simplest mixing length or zero-equation models, one-

equation models and two-equation models, to the most general Reynolds stress turbulence

closures. Eklund et al. [68] simulated subsonic diffusion flames using an algebraic

turbulence model [2]. Due to the limitations of this turbulence model, the effects of

fluctuations of the species concentrations and the temperature, were ignored. Jones

and Whitelaw [69] showed that this underpredicted the extent of combustion. A Direct

Numerical Simulation (DNS) of a reacting mixing layer has been carried out recently by

Givi et al. [70]. The effects of temperature fluctuations was modelled using an assumed

Probability Density Function (PDF) technique, •proposed by Frankel et al. [71]. Narayan

[72], Villasenor et al. [73] as well as Kolbe and Kollmann [74] have applied the two-

equation models for simulating reactingmixing layers. While these are more sophisticated

than algebraic models, they have been developed primarily for incompressible flows,

using a gradient transport hypothesis.

Unfortunately, realistic engineering problems entail "non-gradient" transport [75-78].

This has been observed by l linze [75], for incompressible flows, in measurements of the

wake of an axisymmetric cylinder. In compressible flows, 'counter-gradient" diffusion

• I

has been observed experimentally by Moss [76]. Also, Libby and Bray [77] predicted

that counter-gradient diffusion can occur due to the effect of the mean pressure gradient.

The two-equation model ('k-C:" model) neglects terms involving pressure gradients. In

addition to these shortcomings, the two-equation models cannot predict buoyancy effects

and perform poorly for predicting swirling flows. These defects are eliminated with the

differential Reynolds stress lnodels, as suggested bv Itogg and Leschziner [78].

The combustion of hydrogen and air in the scramjet ..combustor results in absorbing-

emitting gases such as water vapor and hydroxyl radicals. Existence of such gases

i '

[

i,

31

makes the study of thermal radiation from turbulent reacting flows, an important issue.

There are several models available in the literature to represent the absorption-emission

characteristics of molecular gases as described by Mani and Tiwari [22]. Both pseudo-

gray and non-gray gas models have been employed to evaluate the radiative heat transfer

for supersonic combustion. Results of both models were compared and the pseudo-

gray model was found to be computationally efficient. All these studies considered only

laminar flows I22].

tlowever, an important issue for turbulent flames is the effect of turbulence/radiation

interactions [79-85]. The results obtained by Cox [79] shed some light on radiant

heat transfer from turbulent flames. Tamanini [80] obtained numerical solutions for

radiation on turbulent fire plumes. Kabashnikov and Kmit [81] investigated the influence

of turbulent fluctuations on thermal radiation. Experimental data on turbulence/radiation

interactions in diffusion flames were obtained by Gore et al. [82], and Jeng and Faeth [83].

Yuen et al. [8411 discussed non-gray radiation in the presence of turbulence. Fairweather

et al. [85] investigated radiative heat transfer from a turbulent reacting jet.

Second-order turbulence models have been applied recently to reacting as well as

to compressible flows [86-88]. Chen [86] studied subsonic diffusion flames, usingi

a Reynolds Stress model. Farschi [87J investigated heat release effects in reacting

mixing layers, also using a second order model. Sarkar and Balakrishnan [89] applied

a compressibility correction to the e equation, and observed the correct decrease in

growth rate of compressible mixing layers.

As seen from the above literature survey, there is clearly a need for additional

investigation in this key field. The objective of the present study is to investigate

turbulence-chemistry-radiation interactions in supersonic: hydrogen-air diffusion flames.

b[

z_ '

!?

{ •

i

32

The effects of turbulence on the chemistry and radiation heat transfer, are accounted for

by using an assumed Beta-PDF method. The flowfield in the combustor is represented

by the Navier-Stokes equations and by the appropriate species continuity equations. ]'he

finite difference method using the explicit, unsplit MacCormack scheme is used to solve

the governing equations. A truncated 7-step finite rate chemistry model, derived from

the complete mechanism (Table 2.1) is used here. The radiation model used in this study

is the same as discussed in Sec. 2.4 of Chap. 2.

The physical models used in this study are the Beach and Jarrett-Pitz coaxial jets

experiments [89, 90]. A schematic diagram of these coaxial jets experiments is given

in Fig. 4.1, wherein an inner fiJel jet diffuses into an outer air jet. This outer jet is

vitiated with water vapor to enhance the combustion process. The temperature and other

exit conditions for the Beach case are given in Table 4.1. The temperature and other

exit conditions for the Jarret-Pitz experiment are given in "Fable 4.2. Diagnostics for the

Jarrett case could be more reliable, since the Beach experiment is older by a decade.

4.2 Basic Governing Equations

The physical model for analyzing the flowfield in a supersonic combustor is described

by the Navier-Stokes and species continuity equations [1, 2]. Favre-averaging is used

to derive the turbulent flow equations from the Navier-Stokes equations. This is carried

out as

: + ¢' (4.1)

t

where the mean is expressed as

and 4_ denotes ,, _,, T and fi.

= P-__ (4.2)P

i;i-!

" i

plog,_og ]o.f l_!X_OO _q] jo o!]P,m_qn S ['17 "g!_I

_ -'----i

I

i-i

.i

f

Table 4.1 Conditions for the Beach experiment

34

[I 2 Air

Mach No. 2.0 1.9

Temperature 251 K 1495 K

Pressure 0.1 MPa 0.1 MPa

Velocity 2418 m/s 1510 m/s

f-H2 1.0 0.0

t"-O2 0.0 0.241

f-N2 0.0 0.478

f-H20 0.0 0.281

Fuel injector inner diameter, d -- 0.009525 m.

Injector lip thickness = 0.0015 m.

Nozzle diameter, D = 0.0653 m.

lhble 4.2 Conditions for the Jarrett-Pitz experiment

ttydrogen jet (inner) Air jet (outer) Ambient Air

Mach No. 1.0

Temperature 545 K

Velocity 1772 m/s

Pressure 0. I 12 MPa

f-tt2 1.0

t"-O2 0.0

f-N2 0.0

f-It20 0.0

2.02 0.0

1250 K 273 K

1441 m/s 0

0.096 MPa 0.101 MPa

0.0 , 0.0

0.254 0.233

0.572 0.767

0.174 0.0

Fuel injector inner diameter, d = 0.00236 m.

Injector lip thickness = 0.00145 m.

Nozzle diameter, D = 0.01778 m.

The density and pressure are time-averaged and are expressed as

35

ii

p=-_+p

ii

p = _ + P (4.3)

where by definition

p&' = 0 (4.4)

t

.!

,!

and

p = 0

-"77I' = 0 (4.5)

as

A relationship between time-averaged and density-weighted variables can be obtained

-- II _)1

- q_ _ P (4.6)F

This leads to the averaged continuity, momentum and energy equations being expressed

in tensor notation as [88]

op 0(?,_)-- + 0 (4.7)Ot O:r.i

./

f

o(F,7;) o(F_i,_j) 0vOt (";):r.i O:ri

+ (4.8)O:r.i

! ,

and

(9t+

O:rj

-F

O:ri

(r_:/,, j

where the mean heat flux is expressed as

07'

i):r i

and the turbulent energy flux L u can be Obtained as,

E'uli = -CI'] u i + HiltjH i

I t ! III

u ., . u • %-, _ .S-- "+ ' , ._ + _bt'.tl. u'2 }

k= I

The averaged species equations in tensor notation are obtained as

+Ot O._".i c%,j

0 I)

-t- + _t't:O:rj

36

(4.9)

(4.10)

(4.11)

(4.12)

F

In order to close Eqs. (4.7)-(4.12), it is necessao, to provide models or modelled

transport equations for the following quantities •

! I

a. the Reynolds stress tensor u.7;. in Eq. (4.8)t 3

b. the turbulent heat flux 7",' in F.q. (4.11)

c. the turbulent mass flux P""'i " the turbulent temperature flux/3"7" , and the turbulent

species flux p".[_. , all appearing in Eq. (4.6)

d. the turbulent species mass flux @./'£. in Eq. (4.12)1

e. the mean chemical species production term _ in Eq. (4.12).

t

37

II I

Other terms such as ri i uj in Eq. (4.9), are neglected in high Reynolds number turbulence.

c_

i :

4.3 Reynolds Stress Turbulence Model

The exact transport equation for the Reynolds stress ,'._' can be obtained as [75, 88]I .I

+Ol O:r_.

i) iT,.

--P"i_tl" O:v k P_t i_tk i):r),

7'rrm l

(4.13)

where 7'efT, 1 is the production term, i.e.,

F,.i = 7'Crm I

and Term 1I is the viscous dissipation term q.i which is modelled as [88]

'2

: .7 (1+ ,%cii

(4.14)

(4.15)

In Eq. (4.15), the turbulent Mach number is defined as [88]

V 7 I? T(4.16)

?

The term is computed from a transport equation as [88]

Ot + O:r_. _'/,'" ' .p _

-(:'_-PT + i).r_. (4.17)

?_r

]:

?

38

where the model coefficients have the following values •

f',,1 = 1..11, (7,;2 = I..qO, (:, = 0.15 (4.18)

Term II1 in Eq. (4.13) is ttle diffusive transport term l_.it..t, and is modelled as

Tijk.t. =---. ---_ O:r_.

o,;)]-v

where (,._ = 0.018.

(4.19)

The last Tcrm lI" in Eq. (4.13) can be expressed as the stun of a pressure-strain term

llij and a mean pressure gradient term, details of which are given in [88]. The Launder-

Reece-Rodi (LRR) model is applied to the pressure-strain term llij, and is expressed as

H i.i = (',p_'[ _ _¢_i.i - ('3 '_.i- _Pt:t- (4.20)

where (:j = 3.0, Ca = 0.6 and [)i is given by Eqs. (4.13_1.14).

The turbulent heat flux term in Eq. (4.11) is modelled as a gradient transport term,

7',Wi __ (-'_' '" , (4.21)_\7' cg:ri

F_

I ,

i

where (' = 0.0.()..'II.

The turbulent species flux u r _ P _i, the.ij_. in Eq. (4.12), and the turbulent mass flux , t

turbulent temperature flux p_"l '_, as well as the turbulent species flux -" _/, j_,, all appearing

in Eq. (4.6) , are modelled using the gradient transport method from Eq. (4.21), e.g.

i,, .,----_ ( "1"-/)J:_ (4.22)P .It, = c\ O:ri

! •

39

The mean chemical species production term _ in Eq. (4.12), is modelled using an

assumed Beta-PDF for the temperature fluctuations, i.e.,

wt: = w_:. P T'

The Beta-PDF is preferred over other distributions because it closely approximates the

actual DNS simulation of scalar mixing, and is expressed as

= +/.3)k) l'(i_)1'(t_)

(4.24)

where

)O' = To --.._-= -I (4.25a)

\ "r,7

= - ._ -1 (4.25b)

"117

The quantity 7o 2 is the variance of the Beta-Pl)F. detail of which are given in Sec. 5.3,

and I' is the standard gamma function. Since probability space recognizes only values

from 0 to 1, it is necessary to normalize the mean temperature in Eq. (4.25) as

7ql = ? - 7_,,h, (4.26)

n,'lx -- rain

A very similar analysis can be carried out for the Beta PDF for species fluctuations.

4.4 Radiative Interactions

!,

I• ••

dr,[:.

The radiative heat transfer model used in this study is the same as discussed in

Sec. 2.4 of Chap. 2. Therefore. the relevant details need not be repeated here.

In the present study, tlle mean qz_ term in Eq. (4.9) is modeled as

4- o<:,

qR = ./ qR " I'(7") dT (4.27)--00

where the PDF. P(7") is given by Eqs. (4.24)-(4.26). Eqt, ation (4.27) is a new and

simple formulation for the cot,pied turbulence/radiation interactions.

L

!

i •

40

4.5 Method of Solution

The method of solution used in this study is the same as discussed in Sec. 2.5 of

Chap. 2. Additional details are presented in this section. The upper boundary is treated

as a free boundary, wherein the gradients of all variables vanish. The lower boundary

is the centerline of the axisymmetric flow. Consequently, the normal velocity is zero,

on the centerJine. The gradients of all remaining variables on the centerline vanish by

symmetry. Initial conditions are obtained by specifying freestream conditions throughout

the flowfield. An isotropic turbulent shear stress was prescribed as the initial condition

for the Reynolds Stress equations. The resulting set of equations is marched in time,

until convergence is achieved. The details of the radiative flux fornmlation and method

of solution are available in Mani and Tiwari [22].

i _ "

i

i

i

I. •

Chapter 5

RESULTS AND DISCUSSION

The theoretical formulations described in Chaps. 2, 3 and 4 are applied to obtain

results for noneqnilibrium processes in supersonic combustion. The explicit MacCormack

technique has been used to march the governing equations in time, until convergence

is achieved. The two-dimensional Navier-Stokes equations are solved for supersonic

flows with nonequilibrium chemistry and thermodynamics, coupled with radiation, for

hydrogen-air systems. This computer program is an extension of the original SPARK code

[2, 3]. Results for the impact of chemical kinetics on radiative interactions are discussed

first, followed by the results for the interaction of thermochemical nonequilibrium and

radiative heat transfer. Results for the turbulence-chemistry as well as the turbulence-

radiation interactions are presented in the third and fourth parts of this chapter.

5.1 Chemical Nonequilibrium and Radiative Interactions

Studies were conducted to investigate the extent of radiative ,heat transfer in super-

sonic flows undergoing hydrogen-air chemical reactions, using three chemical kinetics

models. These chemistry mechanisms account for an increasing number of reactions

and participating species. For the temperature range considered in this study, the im-

portant radiating species are OHand tt20. The gray gas formulations are based on the

Planck mean absorption coefficient which accounts for detailed information on different

molecular bands. The radiative fluxes have been computed using this "pseudo-gray" for-

T

mulation. The justification for using this model is provide in Mani and Tiwari [22]. The

41

k_

r

42

three chemistry models are obtained from Table 2.1. The 2-step model is given at the

bottom of that table. The first 18 reactions in this table constitute the 18-step model. The

remaining 17 reactions Nos. (19)-(35) complete the 35-step model. Jhe two-dimensional

problem considered in this study is shown in Fig. 2.1.

Figures 5.1-5.4 show the computed results using a 61 × 61 grid, for temperature and

pressure as well as It20 and OII species mass fractions. The oblique shock igniting the

air-fuel mixture, arises from the compression corner at the lower wall (Fig. 2.1). The

hottest regions in the flowfield are in the tipper and lower'wall boundary layers. Figures

5.1 and 5.2 show the effect of the three chemistry models on the temperature and pressure

profiles, varying along x at the location y = 0.02 era. from the lower wall (boundary

layer region). The temperatures in the boundary layer show a gradual increase (Fig. 5.1).

The pressure profiles are plotted at y = 0.13 cm. (inviscid region) and show a sharp

increase due to the shock (Fig. 5.2).

The ignition phase discrepancy of the three chemistry models can be seen in Figs. 5.3

and 5.4. The shock is occurring after x/Lx = 0.3. ttowever, the 2-step model predicts

ignition before the shock (shorter ignition delay) due to the high temperature in the

boundary layer. On the other hand, the 18-step model predicts a longer ignition delay,

I

at x/L× = 0.43 (Fig. 5.3). The 35-step model's prediction of ignition delay appears to be

an average of the other two models. Although the three models do not differ much in in

prediction of temperature and pressure profiles, they do differ significantly in predictions

of species mass fractions (Fig. 5.4).

In order to resolve this discrepancy, a grid sensitivity study was carried out to examine

whether the grid size affects the flow predictions. The results of three grid distributions

31 ×31, 61x61 and 81 ×81 are shown in Fig. 5.5, and it appears that the 61x61 grid is

0

oO

c_

C',l

0

0

0

lJ

,1-1

o

×_

¢J

b-

_b°_

_' aanleaadmal

" _ " : ..... - L : - [ ........ L .....

!

44

, .i

i,

T

! •

! -i "

5

4

3

2

00.0

--_ 2-step

--*-- 18-step

--*- 35-step

!

0.2

Fig. 5.2

I ' I ' I "

0.4 0.6 0.8

X/L xPressure profiles (y=O. 13 cm.)

tt"_

¢) ¢)

r._ t !A oG

C'q ,.. Cr_

O'3¢5

! ° i

o c_

uo!13_:aj ss_m OZH

v,,,,,,m

_G

c_

c;

¢',1

_D

©

©o_

¢DX

t-,

t-.,

©

tt",

t.L

00

!

q p

:!ii

I I

c_

1

c_0

!

0

01.,q

0

×o

E

©

uo[131_aJ ssl_m HO

4._. _L _

J

X .m

,--1 '-2

©

i

i

i:

F

48

sufficient for the present study. The grid points are concentrated in the boundary layer

and shock regions.

The reasons for the varying predictions of species mass fraction by tile three chemistry

models was examined fi_rther and the results are shown in Figs. 5.6 and 5.7. Figure 5.6

shows that tile reaction No. 8 in Table 2.1 is critical in determining the extent of chemical

heat release and II20 production. Reaction No. 8 deals with production of I-tO2 radical.

This reaction is absent from the 2-step model, while it is common to both the 18-step

and 35-step models. Figure 5.6 shows that tile 35-step model experiences nearly a 30%

drop in temperature in tile middle of the channel when the rate of reaction No. 8 is

reduced by a factor of I000 (effectively cutting of the production of the ttO2 radical).

In contrast, the 18-step model shows a 15% drop in temperature, when subjected to the

same reduction in tile rate of reaction No. 8. This shows that the reaction No. 8 controls

the overall t120 production occurring in Table 2.1. Due to the high temperatures (,--,3000

K) in the flowfield, there is a pool of highly reactive free radicals like tt, O, etc. The ttO2

radical is converted to the very reactive O1t radical, by the free radicals (reaction Nos.

11 and 12). This establishes the 1tO2 radical as a very important species in promoting

flame propagation in hydrogen-air flames. A similar reaction sensitivity analysis has been

carried out in [91]. Since tlie 2-step model does not have the HO2 radical, it predicts

lesser amounts of O|t and 1t20.

It was necessary to determine the reason for the higher sensitivity of the 35-step

model to tile HO2 radical, as compared to the 18-step model. Figure 5.7 shows that the

reaction Nos. 21 and 23 in Table 2.1 are critical in determining the extent of chemical

heat release and tt20 production. Reaction Nos. 21 and 23 deal with production of the

NO radical. These reactions are absent from the 2-step and ! 8-step models, whereas they

play an important role in the 35-step model. Figure 5.7 shows that the 35-step model

i,

c,-

i

u

It-,

2

g.E

3500

3000

2500

2000

1500

1000

5OO

0

18-step

Table 2.1

kS/1000

35-step

+

Table 2.1

k8 I1o0o

In

!

0.2

Fig. 5.6

| • | i

0.4 0.6

X/L xEffect of reaction rates

!

0.8

49

i

_+-+:

>

? •

i,

i

i'

b+

t •

(

i +

+-

i •

3500

3000

2500

2000

_. 1500

1000

500

0

50

35-step

Table 2.1

• ,I ooo Ik_2OO

I • i " I I "

0.2 0.4 0.6 0.8 l .0

X/LX

Fig. 5.7 Effect of NO and HO2 reactions

i

i

J

51

undergoes a 10% reduction in temperature, when the rates of reaction Nos. 21 and 23 are

reduced by a factor of 1000 (effectively cutting off the production of the NO radical).

Due to the high temperatures in the flowfield, the usually inert nitrogen dissociates into

the highly reactive N free radical. This free radical N is then oxidized in reaction Nos. 21

and 23, thereby producing tile NO radical. This NO radical converts tile 1102 radical

into the highly reactive OH radical, through reaction No. 29. This confirms that ttle NO

radical is a very important species for flame propagation in a hydrogen-fueled supersonic

combustor. Since the 35-step model has the NO radical, it predicts higher amounts of

Oit and tt20 than the 18-step model.

Based on the above understanding of the chemical kinetics of supersonic hydrogen-

air flames, the radiative interactions were examined. Figure 5.8 shows the profiles of

the normalized streamwise radiative flux qRx predicted by the three chemistry models,

along tile location y = 0.02 cm. from the lower wall. The qRx flUX reduces towards

the end of the channel due to cancellation of fluxes in positive and negative directions.

It is seen from Fig. 5.8 that tile 18-step and 35-step models predict significantly higher

amounts of qRx (50% more and 100% more, respectively) than the 2-step model. This is

because radiative heat transfer is a strong fimction of temperature, pressure and species

concentrations. So, tile larger values of radiative fluxes are caused by higher amounts of

I-t20 concentrations, which in turn, depend on reactions involving HO2 and NO species.

Figure 5.9 shows the variations in the normal radiative flux qr<y along x, at tlle location

y = 0.02 cm. from lhe lower wall. These do not appear to vary significantly between

the three chemistry models, ltowever, in all three cases, the qRy value increases rapidly

after the shock.

Figures 5.10-5.13 show tile computed results for reacting flows with and without

radiation, for the three chemistry models. It is seen that the 2-step model shows only

f,i{:

f

52

Ir

b

10

D

6

4

2

00.0

2-step

-*- 18-step

35-step

• I " !

0.2 0.4 0.6 0.8

X/LxFig. 5.8 Profiles of streamwise radiative flux

,f

t .

53

i,̧

t

J

! •

400

1O00

-----_--'...2._'-"' "-" ....

--_ 2-step ..... --'--

--,.- 18-step

--_ 35-step

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 5.9

X/L xProfiles of normal radiative flux

t/_,

clapom dins- D salgo._d _1meJadtum uo sla_jja uo!m.rpe_l eO['_c ":_!:.,d

x'-I /x

0"[ 8"0 9"0 _'0 g'O 0"0

, I i I i I , I i ' 0

O0.C

000

O00C

000£

OOg£

m

m

u

J

t,/",{lapom dais-81 ) sa[uoJd a.mle.ladtual uo slaajja uo!].,!p.N

×I/X

O'I g'O 9"0 17"0 _'0i I i I i I i I

qo [ "_ "_ !..q

000_

m

m_

_m_¢mmm,m

IroN(

DNId.VIGV_t _ DNI2DV3_I

AqNO DNLLDVg'eI

dins-g[

,.2 .... -_. 2-

if',

(lopom dgls-g_) Salgmd a.m_t_.Iadtum uo sm_jja uo!n:._eN aOI'_c g!Y

Xl/x

O'I 8"0 9"0 17"0 C'O 0"0

i 1 i I , I , I i _ 0

O0_C

0001

O00C

000_

i

'=I.

"=I

t.t%

(IapouJ dins-c) "uJ_ £I'0 = '_ lu s_lljoad _nssaJd uo slo_jja uo!j_..rpu_{ P.I I'_c ?'!'d

×I/X

0[ 80 90 ¢70 _0 00, I i I , I i 1 i 0

_._L.L,._..__.=_

ONI.I.VIGV_ _ DN_V_

XqNO ON_V_8

¢.,

Ct,.,

17

i

w

i i¸7-¸..... :2 ..: 21 ! -_2_-.

£

he',l_pom d_ls-s[ ) "tuz £I'0 = '( m s_IgoJd _anss_ad uo s_z_j.j, uo!m!pe,_I qI I'; "_!-5

×I/X

0"_ 8"0 9"0 17"0 _'0 O'O, I i I , I , I , 0

ONIX,VICI'v'd :'g DNILDV_:I

AqNO DNLLDV_'d

ci_ls-81

C

C

17

12pom dols-s£) "m_ ci" 0 - X _13s_lgoJd olnssoJd uo s_a_j._ uo!_!p_t ,3[ [c "_!_

×"I/X

0"I 8"0 9'0 _'0 _'0 0"0

; i 1 i I i I I I a 0

ONI.LVICIV_t ,_ ONId.DV3_I

A_INO ONLLDV_t_I

d_ls-_c

[

m

c

#

./. i ¸

(Iapom dins- D Salgo.]d OtH uo .sln_j_ uo!ln.rp,_l _,ZI_C _!H

×"I / X

0"[ 8"0 9"0 17"0 ['0 0"0

l

¢-

m

J

mw

z._L.;........ : + _+ _a +_:.; ....... n ...... - ...... ___ ...... a ............... . .... "........ 5 .+.a+ + .2_ ._2, £ ..... :

i=POm d_ls-si ) S_luoJd O;H uo sl=_jj_ uo!le.!pe_t q_I'g g!H

x_I/x

0[ 8"0 9"0 t7"0 _0

DNI.I.VICIV_t _2 DNI.IOV_t'd --_

XqNO DNI2OV_t

i

d_!s-81

mm

N

m

.2..£_2L 2_ ...... '; .... £ £i....... _22..L_]:

6",1

(lapotu dals-_c) salgmd O_-H uo slaajja uope..rpe,_I agI',c _!g

×"I/X

O'I 8"0 9"0 17"0 ;'0

DNId,VIG'v"8 _ ONI,LDV_t_t

XqNO ONI.LD'v'=j_t

dms-cc

1:'0

i;0

_o

UAom;9

I.

r_

m

M

m

0.04

0.03

2-step-..o- REACTING ONLY

REACTING & RADIATING

I I I

0.0 0.2 0.4 0.6 0.8 1.0

X/L xFig. 5.13a Radiation effects on OH profiles (2-step model)

........... - ,--; .......... : ,: _, = - .-................. _ 7-:........ _ .....:...... ,:-_ _----_-_i_ ....:_:

Jmu

om

W

J

0.04

0.03

0.02

0.01

0.00

i

18-step..-o- REACTING ONLY

REACTING & RADIATING

!

0.6

X/L x

I

0.8

Fig. 5.13b Radiation effects oil OH profiles (18-step model)

tg",

0"12 8"0 9"0 17"0 ;'0 0000"0

DNI.I.VI(:IV_I _ DNI.LDV:g'd -_

AqNO DNLLDV!t'8 --.-o-

dms-_;c

u

r_r_

mllll o

m

i ¸,

Kt

66

a slight effect of radiative interaction, as cornpared to the 18-step and 35-step models.

The 18- and 35-step models, with radiative interaction, predict lower temperature and

lower H20 and OH concentrations after tile shock. This is because of tile qRx flux which

reduces the total energy. For reacting flows without radiation, it was seen earlier from

Figs. 5.3 and 5.4 that tile l g-step model had a longer ignition delay (ignition at x/Lx

-- 0.43), while the 35-step model had a shorter ignition delay (ignition at x/Lx = 0.38).

Another effect of radiative interactions, seen in Fig. 5.12, is to nullify this difference in

predictions of ignition delay. For both l g-step and 35-step models, with radiation, the

shift in ignition is seen to occur by the same amount, x/lJx = 0.05. No such effect is seen

on the ignition characteristics of the 2-step model. The results of Figs. 5.10-5.13 show

the "cooling effect" of radiative interactions. The heat release from exothermic chemical

reactions usually thickens the boundary layer, but the lower temperatures produced by the

radiative interactions oppose this nonequilibrium effect. Since the ignition temperature

is lowered, the ignition delay is also affected by the radiative heat transfer.

5.2 Thermochemical Noneq||ilibrium and Radiative Interactions

Studies were conducted to investigate the extent of radiative heat transfer in su-

personic reacting flows undergoing vibrational relaxation. For ,the temperature range

considered in this study, the important radiating species are OIt and I-t20. The LTE

(local thermodynamic equilibrium) radiative fluxes have been computed using the same

formulation used in Sec. 5. ! of this chapter. The non-LTE radiative fluxes are calculated

using a new approach discussed in Chap. 3. In order to avoid expensive computer us-

age, tile chemistry model used in this study is a truncated 7-species, 7-step mechanism,

derived from the first 7 reactions in Table 2.1. The five chemical species undergoing

dL

f

i !

L

r ¸

i

i

! i

i_

f

i

i

67

vibrational relaxation are t12, O2, I!20, Oit and N2. Tile monatomic species t1 and O do

not subscribe to the "dumbbell" theory of harmonic oscillators, and so are in equilibrium.

In order to validate the theoretical formulations discussed in Chap. 3, comparisons

were made with results fi'om a hypersonic workshop [53]. The physical model for this

study is shown in Fig. 3.1. Even though the present study is for 1-12-- air chemistt3',

and the results presented in [53] are for N2 -- 02 reactions, this was the best comparison

available. Figure 5.14 shows the one-dimensional results for translational and vibrational

temperature obtained for air chemistry in an expanding nozzle. A total of 101 grid points

were used in the flow direction. It can be seen from Fig. 5.14 that the thermochemical

nonequilibrium model used in the present study, compares quite favorably with the one

used in [53]. The slight discrepancy in prediction of vibrational temperature Tv is due

to differences in the method of computing the relaxation time. In the present work, the

relaxation time is calculated using a relation given by Millikan [63], whereas the method

used in [53] is based on a relation given by Vincenti and Kruger [24].

The physical model used for the remaining part of this study is tile nozzle and

is shown in Fig. 3.2 The first step was to assume chenlical nonequilibrium (CNE) in

all cases. Figure 5.15 shows the one-dimensional results, using 101 grid points, for

!

the temperature and pressure variations along x. The temperatures exhibit relaxation

along tile nozzle (Fig. 5.15a). The vibrational temperature Tv is shown for the species

(H20) that exhibits strongest nonequilibrium effect, and it deviates significantly from the

translational-rotational temperature T. This shows that thermochemical nonequilibrium

(TCNE) is still present in the nozzle, and reduces the translational temperature. The

pressure profiles (Fig. 5.15b) do not show any effect of thermal nonequilibrium.

r

b,.

6000

5000

4000

3000

2060

1000

00.0

Tv - present

T- present

Tv Ref.[53]

T Ref.[53]

x

n [] 0

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Fig. 5.14 Air chemisn'y temperature profiles (l-D)

(G-I) Salyoad aanlgaadm_L n_cIc _!_t

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71

Based on the above understanding of thermochemical nonequilibrium in supersonic

hydrogen-air flames, the radiative interactions were examined for a two-dimensional flow.

A I01 ×31 grid was used for this part of the study. Only the upper half of the flow was

computed, since the nozzle is symmetric about the centerline. The results were plotted

after every four grid points. Three y locations were considered, viz. j = 1, j --jmid,

and j = jmax-l, corresponding to the centerline, midway between centerline and wail,

and wall boundary layer, respectively. The local tilermodynamic equilibrium (I3E) and

non-IJ'E results were obtained by using a value of 7! = ltc/qr = 0.0 and 5.0 respectively,

for the nonequilibrium parameter in Eq. (3.13).

Figure 5.16a shows profiles of streamwise radiative flux using 101 ×31 and 101 ×51

grids, for the purpose of a grid resolution study. It appears that the 101 ×31 grid used

in the present study is sufficient. Figure 5.16b shows the profiles of the normalized

streamwise radiative flux qRx along two y locations. It should be noted that the radiative

flux at the centerline j = I, is negligible, and hence is not plotted. It can be seen that the

ql_,x flux in the wall boundary layer j = jmax-l, (Fig. 5.16c) is higher than at the other

two locations. This is due to the adiabatic wall boundary condition, which precludes

any heat transfer to or from the wall. An important effect of thermal nonequilibrium is

to reduce the radiative interactions. The qRx decreases towards'the nozzle exit due to

cancellation of fluxes in the positive and negative directions.

Figure 5.17 show's the variations of the normal radiative flux qRy along x, at two y

locations, ttere also, the radiative flux at the centerline j = 1, is zero (because of the

symmetry boundary condition) and is not plotted. It can be seen that the qRy flux increases

only slightly in the positive y direction, reaching a maximum in the wall boundary layer

(Fig. 5.17b). This is because of the optically thin assumption, which means that there is

negligible loss of radiative flux from the wall to neighboring gas molecules. Also, thermal

t_q

cil-x_;tu[ = D llg,_ n3 _ xn[4 _^!lut.pgJ a._t,_u.l_.Ils uo ,(.pros uo!lnlos_J PUO

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nonequilibrium reduces the radiative interactions. The qR:, profiles exhibit another peak

near the nozzle inlet, because of sudden increase in radiating species due to chemical

reactions.

Figure 5.18 shows the temperature profiles along the three y locations. It can be

seen that vibrational nonequilibrium reduces the translational-rotational temperature. The

radiative interactions serve to negate this thermal nonequilibrium effect (especially in tile

wall boundary layer, Fig. 5.18c). The oscillations in tile temperature profiles (Figs. 5.18a

and 5.18b), occur only in the presence of radiative interactions. These oscillations are

due to assumption of optically thin radiation, wherein there is a negligible loss of qR to

the wall from the gas molecules, it is interesting to see that this numerical disturbance

is absent at the wall (Fig. 5.18c). It can be seen that the temperature at the midway (i =

jmid) location is higher than at the other two locations. This is because of the heat release

due to chemical reaction. The temperature in the wall boundary layer (j = jmax-1) as seen

in Fig. 5.18c is lower than the centerline temperature. This is because of the adiabatic

wall boundary condition, which prevents heat transfer outside the wall. Consequently,

the wall temperature rises towards the nozzle exit.

Figure 5.19 shows pressure profiles along the x direction. The pressure oscillations

(Figs. 5.19a and 5.19b) occur only in the presence of radiativd interactions. This is

because of the optically thin assumption, which means that there is negligible loss of

radiative flux from the wall to neighboring gas molecules. A reduction of pressure due

to vibrational nonequilibrium can be seen. A trend similar to the temperature profiles

(Fig. 5.18) is observed. This is analogous to the thickening of the boundary layer on a

fiat plate (i.e. lowering of the pressure) in tile presence of thermal nonequilibrium.

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Figures 5.20 shows variations of the vibrational temperature at three y locations. An

interesting effect of radiative interactions is to reduce the vibrational temperature, thereby

negating the effect of thermal nonequilibrium. A similar observation has been made in

[67]. The reduction in vibrational temperature is due to the qr_x flux which reduces the

total energy.

Figure 5.21 shows profiles of water mass fraction at three y locations. They follow

a pattern similar to the temperature and pressure profiles (Figs. 5.18 and 5.19). The

peak water production is found to occur at x/Lx = 0.05. Thus, it can be seen that

the nonequilibrium parameter in Eq. (3.13) serves to illustrate the relative importance

of vibrational relaxation (collision process) over radiative relaxation (emission process).

The non-LTE process is emission dominated. On the other hand, the LTE process is

collision dominated.

5.3 Turbulence-Chemistry Interactions

Studies were conducted to investigate the extent of turbulence-chemistry interactions

in supersonic flows undergoing hydrogen-air chemical reactions. The SPARK code was

modified to include a Reynolds stress turbulence model [89]. The essential modifications

in the present work result in the program's capability to compute a xisymmetric flows and

nonpremixed hydrogen-air combustion. Furthermore, a Beta-PDF has been incorporated

into the computer code. In order to avoid expensive computer usage, the chemistry

model used in this study is a truncated 7-species, 7-step mechanism derived from the

first seven reactions in Table 2.1. The resulting formulation is validated by comparison

with experimental data on reacting supersonic axisymmetric jets. The physical models

considered for this study, deal with the nonpremixed combustion of supersonic coaxial

jets. These are the Beach experiment [89] and the Jarrett-Pitz experiment [90]. Firstly,

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(p!m[ = .1")saJnmJadtum V,uo!l_Jq!A qo_'_c "_!H

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l ' l ' I

0.4 X/L0.6 0.8Water mass fraction profiles (j = jmax-l)

1.(

i •

IU"

91

the computed results for the Beach case will be discussed, and this will be followed by

the results for the Jarret-Pitz case.

i" i

! :

Figure 4.1 describes the physical model for the Beach and Jarrett-Pitz experiments.

The temperature and other exit conditions for the Beach experiment [89] are given in

Table 4.1. The outside diameter of the fuel nozzle (d = 0.009525 m.) is used to normalize

the axial and radial profiles. The length of the flow domain is taken to be 28 diameters,

i.e., x = 0.2667 m. Initial profiles for tile flow variables are obtained by computing flat

plate solutions. This is carried out separately for tile filel and air streams and then the

two flows are combined. This approach is better than the "ad-hoc" initial profiles chosen

by several authors [86, 87]. Figure 5.22 shows the initial temperature profile in the radial

direction. The peak temperature is a result of assuming a constant wall temperature of

1700 K for the flat plate flows.

Figure 5.23a shows the density profiles using 61 × 61 and 81 x 81 grids, for the purpose

of a grid resolution study. It appears that the 61 ×61 grid is sufficient for the present

study. Figure 5.23b shows radial profiles of the major species concentrations at an axial

location of x/d = 8.26. The computed results are compared with experimental data and

the t12 and N2 profiles show reasonably good comparison at all radial locations. A good

match for the O2 profiles can be seen at locations greater than or equal to r/d = 0.6. This

is also the case with the profiles of t120 mass fraction, where the peak of the flame in the

central jet can be observed. Discrepancies between computed and experimental results

can be attributed to inadequate predictions of turbulent mixing and initial conditions.

In order to investigate tile extent of turbulent mixing, it is possible to examine a

mixing scalar, known as the mixture fraction .f, which is defined as the normalized mass

fraction of an atomic element originating from one of the input streams, viz. usually tile

i

92

f

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!.,

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[,..

2000

1600

1200

800

400

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0.20 0.40 0.60 0.80 1.00

r/dFig. 5.22

• ! • !

1.20 1.40

Radial profile of initial temperature

!

1.60

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I)9"1 I)l;"l 0_l1 J ! J 1

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fuel stream. The variable f can only vary from zero to one for a two-stream flow. Under

the assumption of equal diffusivities of all species, the mixture fraction is independent of

the progress of the chemical reaction. This is an important characteristic of the mixture

fraction and serves to indicate the extent of mixing. Experimental data for the mixture

fraction are deduced from the measured mass fractions of the maior species, using I| as

the conserved element, i.e.,

.7 2

f = .fH2 + -(_.fH20 -- Y_.fH20 (5.1), '2 oo

.f It2 _.f H2o

where superscripts * and _ denote the fuel stream and air stream at the nozzle exit,

respectively. A similar analysis has been presented in [73]. Figure 5.24 shows predicted

and experimental mixture fraction profiles in the radial direction, at the location x/d =

8.26. It can be seen that there is reasonable agreement between the computations and the

experiment, except near the centerline r/d = 0.0. This implies that the turbulent transport

model is satisfactorily predicting the extent of turbulent mixing for regions away from

the centerline.

Figure 5.25 shows the predicted radial profiles of the minor species mass fractions.

The presence of non-negligible amounts of fi'ee radicals denotes the extent of chemical

nonequilibrium. Also, higher amounts of O and O1t radicals are formed relative to the

tt radicals.

r

i .

Figure 5.26 shows the predicted density profile in the radial direction, at the same

location x/d -- 8.26. It can be seen that the density of the inner (fuel)jet is lower than

the outer (air) jet. ttowever, the density decreases in the vicinity of the central jet (r/d =

0.4 I 0.5). This is due to the heat release from the chemical reactions. Figures 5.27a-b

show the radial profile of the normalized turbulent shear stress at the axial location x/d

= 8.26. It is interesting to observe a localized reduction of turbulent shear stress in

96

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ii

¢,,}

IN

IlkI,,,,

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1.0

0.8

0.6

0.4

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0.0

0.0O{} 0.200 0.400 0.600 0._}(}

r/d

present

Experiment

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Fig. 5.24 Profiles of mixture fraction

i

'J'l

_°

,...a

_°

_°

Mass fr_etion

t'_J

c_

C_

I ,

0_0

0

---.1

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R.S. model

Mxg. lth. model

• !

0.60

m

m

m

m

m

m

m

!

1.20

| , !

0.80 1.00 1.40r/d

Radial profile of normalized turbulent shear stress

m • m mm• ! ' T

1.60

0

÷

2_1 )/A,n-

I01

i

fi::i

t

i i

t

t

! :'

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Fig.

0.200

5.27c

llll

zero value

zero gradient

Experiment

ill

0.400 0.600 0.800 1.000 1.200 1.400 1.600

rid

Mixture fraction profile -- effect of different shear stress B.C.'s

_T

L

i?

L

iL

i

IL•

102

the same region as the density (Fig. 5.26). This is consistent with the Prandtl's mixing

length model

011 - ()ll

T = t,, = pl-' (5.2)

where 1 is the mixing length. Equation (5.2) can be expressed as

Equation (5.3) suggests that if lhe mean velocity profile is not strongly altered by heat

release, then the turbulent shear stress will decrease with decreasing density. A similar

argument has been presented in [92]. Figure 5.27b illustrates the turbulent shear stress

profiles using two different boundary conditions, viz. zero gradients or zero value at the

boundaries. Figure 5.27c shows that this change in the shear stress boundary condition,

does not produce any appreciable change in the mixture fraction.

The schematic diagram for the Jarrett-Pitz experiment [90] is also given by Fig. 4.1.

The temperature and other exit conditions for this experinaent are given in Table 4.2.

The outside diameter of the nozzle (D = 0.01778 m.) is used to normalize the axial

and radial profiles. The length of the flow domain is taken to be 5.7 diameters, i.e., x

= 0.1016 m. Initial profiles for the flow variables are obtained by computing flat plate

solutions. This is carried out separately for the fuel and air streams and then the two

flows are combined. A fixed location of x/D = 0.85 is chosen since experimental data

are available for CFD code validation.

The mixture fraction is defined in the same manner as in the Beach experiment [89].

Figure 5.28 shows predicted and experimental mixture fraction profiles in the radial

direction, at the location x/D = 0.85. It can be seen that there is reasonable agreement

!7

J

103

,!

J

:i

1.0

0.8

0.6

0._

0.2

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2

, - * • _ : I' !_1, , : , . _ ,;

0 _ 04 0.6 08- " r/D "

Fig. 5.28 Radial profile of mixture fraction

i ¸ ,

104

between the computations and the experiment. This implies that the turbulent transport

model is satisfactorily predict tile extent of turbulent mixing.

Figure 5.29 shows the predicted and experimental radial profiles of the temperature.

The presence of a central peak denotes the main mixing and combustion zone. It can

be seen that this peak near 1500 K is predicted correctly. Discrepancy in the location

of this peak can be attributed to non-symmetric experimental results. Similar shifting of

temperature peaks has been observed earlier I72].

Figure 5.30 shows the radial profile of the water mass fraction, viz. the flame. A

peak [t20 mass fraction is observed at the same location as the temperature peak. A

more diffuse profile after r/I) = 0.6 could be achieved, if some OIt radical seeding [_

!-3%] were carried out. This would alter the ignition profile, as discussed in Sec. 5.1.

Figure 5.31 shows the radial profile of the normalized turbulent shear stress at the

same axial location x/I) = 0.85. It is again interesting to observe a localized reduction of

turbulent shear stress in the same region as the flame. This is because of the reduction

in density due to heat release.

Figure 5.32 shows the radial profile of the normalized right hand side (R.H.S.) of the

turbulent shear Reynolds Stress equation, Eq. (13). It is interesting to note that the initially

isotropic (i.e. initial R.tt.S. = 0.0) turbulent shear stress undergoes "nonequilibrium" in

the region of the diffusion flame. This is because, in the flame region, the R.It.S. is non-

zero. Away from the flame, the turbulent shear stress "returns-to-isotropy". Therefore,

this can be termed as a "relaxation" process.

Figures 5.33-5.34 show the effects of three different "pressure-strain" models on the

flow characteristics. These models are -- LRR (Launder, Reece and Rodi), SSG (Sarkar,

Speziale and Gatski), and S-L (Shih-Lumley). Details of the pressure-strain models are

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0.] j

f0.0 ----: -

0.0

_t

J

I1:

f:g

/X

0.2 0.4 0.6 0.8r/D

Fig. 5.30 Profile of water mass fraction

oO

|

',,0

0

|

0

0

I

O400

0

0-I

I=

°_

0

I=

0

m

0

m

do°_

i._

P-_I) I,*,,n"

I ' i ' l ' i ' I

c-I ,--, _ 0 0

c{ c; c_ d

' I

0 "<I-0 0

d oI

c'l

0

",2)

0

C"_

0

0

co °0

0I

0o_

0

m

0lu

P..

t",l

d0o_

[.m

..._5[,

._z:(;'_°FI - l_"ffl) / "S'H'Hr_

109

i

i _

D

!

0.008

0.006

0.004

0.002

0.0000.O00

m LRR

SSG

x S-L

0.200 0.4(}0 (}.6(}(}

r/D

Fig. 5.33 Effect of "pressure-strain" model on turbulent shear stress

0.800 1.0(}0 1.2(}(}

f

II

110

i "

i ;

f_

b

i,q

i ,

I.,

I-

2000

1600

1200

800

400

LLR

_' SSG

x S-L

1 I l I I I

0.200 0.400 0.600 0.800 1.000 1.2(.)0

Fig. 5.34

r/DEffect of "pressure-stalin" model on lemperamre

.

I

i

iL,

111

available in [93]. Figure 5.33 shows that there is no difference between the effects of the

three pressure-strain models on the normalized turbulent shear stress. Figure 5.34 also

shows no difference in the temperature predictions. This is because the flow considered in

this study does not have very strong anisotropies, and is mainly combustion-dominated.

A similar argument regarding "pressure-strain" models for diffusion flames has been

presented in I87]. ttowever other kinds of flows, e.g. swirling combustion, could offer

better opportunities for testing these "pressure-strain'" models.

Figure 5.35 shows the II20 mass fraction profile, based a preliminary study conducted

on the effect of a multivariate Beta PDF for species fluctuations. Details of this PDF are

given in [94]. A special case of the mixing of two scalars was conducted. The N2 species

mass fraction was one scalar and the sum of the remaining species mass fractions was

the second scalar. It can be seen from Fig. 5.35 that the species PDF in the present form

does not show an effect on the tt20 mass fraction. This is because individual reactions in

the chemistry models respond differently to the PDF. Some reaction rates are accelerated,

while others are slowed down. Furthermore, some critical reactions are absent from the

7-step model used in this study. A sensitivity analysis for hydrogen-air chemical reactions

has been discussed earlier in Sec. 5.1 of this chapter. In addition, chemical equilibrium

has likely been reached (reactions have run to completion), implying that changes to the

R.tt.S. of Eq. (4.12) will not produce significant effects. This is very similar to the result

shown in Fig. 5.14, wherein two different methods have been used for computing the

vibrational relaxation time. Despite this, the final equilibrium value of Tv, is nearly the

same in both cases.

!12

( •). :

,+ .

ZI

0.5

,. 0.4U

om

t

m 0.3t_

©N 0.2

0.1

0.0

X

with species PDF

No species PDF

JI i

0.0 0.2 0.4 0.6 0.8 1.0 1.2

r/D

Fig. 5.35 Effect of multivariate species PDF on water mass fraction

5.4 Turbulence-Radiation Interactions

113

r

i

L

r

Studies were conducted to investigate tile extent of radiative heat transfer in su-

personic turbulent reacting flows undergoing hydrogen-air chemical reactions. For the

temperature range considered in this study, the important radiating species are OH and

H20. The turbulence models used in this study are tile same as in Sec. 5.2 of this chapter

-- tile turbulence is accounted for via a Reynolds Stress model and tile temperature fluc-

tuations are modeled with a Beta-PDF. The radiative interactions have been computed

using tile same formulation used in Sec. 5.1 of this chapter -- the radiative heat transfer

is simulated with a tangent slab model employing tile pseudo-gray formulation. 7he

chemistry model used in this study is a truncated 7-species, 7-step mechanism, derived

front the first seven reactions in Table 2.1. Tile resulting formulation is a simple ex-

tension of Sec. 5.2, and tile physical model used for the present work is the Jarrett-Pitz

experiment [90].

A schematic diagram of this experiment is given in Fig. 4.1. The temperature and

other exit conditions for tile nozzle are given in Table 4.2. The outside diameter of tile

nozzle (D = 0.01778 m.) is used to normalize the axial and radial profiles, The length of

the flow domain is taken to be 5.7 diameters, i.e. x = 0.1016 m. Initial profiles for the

flow variables are obtained by computing fiat plate solutions. This is carried out separately

for the fuel and air streams and then tile two flows are combined. A fixed location of

x/D : 0.85 is chosen since experimental data are available for CFD code validation.

Figure 5.36 shows radial profiles of the streamwise radiative flux qRx. Two interesting

phenomena can be observed here. Firstly, the radiative flux increases in the region of the

flame. This is because radiative heat transfer is a strong function of temperature and 1,,t20

mass fraction. Secondly, tile radiative heat transfer is enhanced by accounting for the

m

o,,i0

|

Ci ×g_

N

o_

|

IX

E

o

C',I

0

115

turbulent fluctuations in temperature. In this case, a fixed value of 0.4 for the variance

of the temperature fluctuation is assumed, i.e. 7_-/1 = 40 %. This parametric approach

is preferred over solving a "g-equation" for tile temperature variance. This coupling of

turbulence and radiation is achieved via a Beta PDF for temperature.

fi

/

Figure 5.37 shows that tile effect of radiative heat transfer is to lower the temperature.

This is the "radiative cooling effect" and is strongest only in tile flame region.

i.

116

i

{

(+,:

t_

2000

1600

1200

800

-----ill-- withoul ra(iiati(_il

with radiation

400

i

r

00.0 0.2 0.4 0.6 0.8 1.0 1.2

r/D

Fig. 5.37 Effect of "turbulent/radiation" coupling on temperature

i

I:

I,

Chapter 6

CONCLUSIONS

This study presents a systematic investigation of nonequilibrium processes in super-

sonic combustion. The two-dimensional, elliptic Navier-Stokes equations were used to

investigate supersonic flows with nonequilibritun chemistry and thermodynamics, cou-

pled with radiation, for hydrogen-air systems. The explicit, unsplit MacCormack finite-

difference scheme was used to advance the governing equations in time, until convergence

was achieved. The chemistry source term in the species equation was treated implicitly

to alleviate the stiffness associated with fast reactions. Specific conclusions of studies

conducted on premixed and non-premixed flows are presented here briefly.

Results obtained for the first part of this study indicate the radiative interactions

varied substantially, depending on reactions involving I102 and NO species, and that this

could have a noticeable influence on the flowfield. Also, it is observed that the difference

in the ignition delays of two chemistry models involving HO2 reactions is nullified as a

result of radiative interaction. The results also showed that the streamwise radiative flux

reduces the temperature and concentration of the species. This effect is a strong function

of the amount of 1t20 species concentration.

Results obtained for the second part of this investigation show that the presence of

nonequilibrium in the expansion region of the nozzle. This reduces the temperature,

pressure, species mass fractions as well as the radiative fluxes. The effect of radiative

117

i ,

i,il

!18

interactions is to reduce the extent of thermal nonequilibrium due to additional mode of

energy transfer.

Results obtained for the third part of this study indicate that the effect of heat release

is to lower the turbulent shear stress and the mean density. Also, it is noted that the

production of turbulence is a "nonequilibrium" process. Results obtained for the last

part of this investigation show that the effect of turbulence/radiation interactions is to

enhance the radiative heat transfer.

Based on the present study, several recommendations concerning the extensions of

this work are suggested. Effects of ionization should be included in the thermal nonequi-

librium investigations. Also, effects of anharmonic oscillators should be considered in

the latter studies. Nongray radiation heat transfer should be investigated and models for

multi-dimensional radiation should be developed. Extensions to other combustion prob-

lems should be carried out. Last but not in the least, effect of species PDF on radiative

interactions should be studied.

119

i /

L

!"

r_

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in the Computations of Plane Channel Flows," ICASE Report No. 92-19, April 1992.

94. Girimaii, S. S., "A Simple Recipe for Modeling Reaction Rates in Flows with

Turbulent Combustion," AIAA Paper 91-1792, June 1991.

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AUTOBIOGRAPHICAL STATEMENT

Rajnish Chandrasekhar was born in Bangalore, India on August 16, 1962. He

received his undergraduate Honors degree in Aeronautical Engineering from the Indian

Institute of Technology at Kharagpur, India in May 1985. Subsequently, he obtained his

Master's in Mechanical Engineering from Concordia, University in Montreal, Canada in

May 1988. His Master's research was funded by the National Sciences and Engineering

Research Council (NSERC) of Canada as a Research Assistantship.

He is a student member of the American Institute of Aeronautics and Astronautics

(AIAA), and the American Institute of Chemical Engineers (AIChE).

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